{"id":6798,"date":"2021-12-30T21:44:51","date_gmt":"2021-12-30T21:44:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6798"},"modified":"2026-02-17T21:20:46","modified_gmt":"2026-02-17T21:20:46","slug":"1-12-logical-arguments","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-12-logical-arguments\/","title":{"raw":"1.12 Logical Arguments","rendered":"1.12 Logical Arguments"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>What is a logical argument?<\/li>\r\n \t<li>Standard Forms of Arguments\r\n<ul>\r\n \t<li>Law of Detachment<\/li>\r\n \t<li>Law of Contraposition<\/li>\r\n \t<li>Law of Syllogism<\/li>\r\n \t<li>Disjunctive Syllogism<\/li>\r\n \t<li>Fallacy of the Converse<\/li>\r\n \t<li>Fallacy of the Inverse<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nAll of the work we have done so far in building the basics of logic has prepared us for the real point which is analyzing logical arguments logically.\u00a0 We want to use logic to evaluate an argument or situation and decide what is and is not reasonable.\u00a0 Setting up our truth tables with letters representing the simple statements allows us to check our opinions and emotions at the door and just focus on the operations.\r\n\r\nA logical argument consists of two parts:\u00a0 a set of premises and a conclusion based on the premises.\u00a0 Our goal is to decide whether an argument is valid or invalid.\u00a0 The premises are supporting evidence for the conclusion.\u00a0 Logic is not about deciding if something is true.\u00a0 Logic is about deciding if the conclusion can be deduced by the given premises.\r\n<div class=\"textbox\">\r\n<h3>Valid Argument<\/h3>\r\nAn argument is <strong>valid<\/strong> if the conclusion follows from the premises.\u00a0 Otherwise, an argument is <strong>invalid<\/strong>.\u00a0 To be <strong>valid<\/strong>, the conclusion of an argument has to follow from the premises.\u00a0 Reasoning that leads to an invalid argument is called a <strong>fallacy<\/strong>.\r\n\r\n<\/div>\r\nThe method that we will use in this class to determine the validity of an argument is by using a truth table.\r\n<div class=\"textbox\">\r\n<h3>Analyzing arguments using truth tables<\/h3>\r\nTo analyze an argument with a truth table:\r\n<ol>\r\n \t<li>Represent each of the premises symbolically<\/li>\r\n \t<li>Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.<\/li>\r\n \t<li>Create a truth table for that statement. If it is always true, then the argument is valid.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the argument:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If you bought bread, then you went to the store<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>You bought bread<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>You went to the store<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"23681\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23681\"]\r\n\r\nWhile this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.\r\n\r\nWe\u2019ll get B represent \u201cyou bought bread\u201d and S represent \u201cyou went to the store\u201d. Then the argument becomes:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]B[\/latex]<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\r\n<td>[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the truth table for [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]\u00a0is always true, this is a valid argument.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the mall, then I\u2019ll buy new jeans.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I buy new jeans, I\u2019ll buy a shirt to go with it.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I got to the mall, I\u2019ll buy a shirt.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet <em>M<\/em> = I go to the mall, <em>J <\/em>= I buy jeans, and <em>S<\/em> = I buy a shirt.\r\n\r\nThe premises and conclusion can be stated as:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can construct a truth table for [latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]M[\/latex]<\/td>\r\n<td>[latex]J[\/latex]<\/td>\r\n<td>[latex]S[\/latex]<\/td>\r\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\r\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)[\/latex]<\/td>\r\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\r\n<td>[latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the truth table, we can see this is a valid argument.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>Forms of Valid Arguments<\/h3>\r\nRather than making a truth table for every argument, we may be able to recognize certain common forms of arguments that are valid (or invalid). If we can determine that an argument fits one of the common forms, we can immediately state whether it is valid or invalid.\r\n\r\nLaw of Syllogism\r\n\r\nLaw of Detachment\r\n\r\nLaw of Contraposition\r\n\r\nLaw of Disjunctive Syllogism\r\n\r\n<\/div>\r\nThe previous problem is an example of a syllogism.\r\n<div class=\"textbox\">\r\n<h3>Law of Syllogism<\/h3>\r\nA syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]p{\\rightarrow}q[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]q{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]p{\\rightarrow}r[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis is sometimes called the transitive property for implication.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>more on the transitive property<\/h3>\r\nThe transitive property appears regularly in the various branches of mathematical study. For example, the transitive property of equality states\r\n\r\nif [latex]a = b[\/latex] and [latex]b = c[\/latex] then [latex]a = c[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I work hard, I\u2019ll get a raise.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I get a raise, I\u2019ll buy a boat.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I don\u2019t buy a boat, I must not have worked hard.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"880229\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"880229\"]\r\n\r\nIf we let <em>W<\/em> = working hard, <em>R<\/em> = getting a raise, and <em>B<\/em> = buying a boat, then we can represent our argument symbolically:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]H{\\rightarrow}R[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>[latex]R{\\rightarrow}B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>[latex]\\sim{B}{\\rightarrow}{\\sim}H[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe could construct a truth table for this argument, but instead, we will use the notation of the contrapositive we learned earlier to note that the implication [latex]{\\sim}B{\\rightarrow}{\\sim}H[\/latex]\u00a0is equivalent to the implication [latex]H{\\rightarrow}B[\/latex]. Rewritten, we can see that this conclusion is indeed a logical syllogism derived from the premises.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nIs this argument valid?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the party, I\u2019ll be really tired tomorrow.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Premise:<\/td>\r\n<td>If I go to the party, I\u2019ll get to see friends.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Conclusion:<\/td>\r\n<td>If I don\u2019t see friends, I won\u2019t be tired tomorrow.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[ohm_question]25956[\/ohm_question]\r\n\r\n<\/div>\r\nLewis Carroll, author of <em>Alice in Wonderland<\/em>, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve the puzzle. In other words, find a logical conclusion from these premises.\r\n\r\nAll babies are illogical.\r\n\r\nNobody who can manage a crocodile is despised.\r\n\r\nIllogical persons are despised.\r\n[reveal-answer q=\"814448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814448\"]\r\n\r\nLet B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.\r\n\r\nThen we can write the premises as:\r\n<p style=\"text-align: center;\">[latex]B{\\rightarrow}I\\\\M{\\rightarrow}{\\sim}D\\\\I{\\rightarrow}D[\/latex]<\/p>\r\nFrom the first and third premises, we can conclude that [latex]B{\\rightarrow}D[\/latex]; that babies are despised.\r\n\r\nUsing the contrapositive of the second premised, [latex]D{\\rightarrow}{\\sim}M[\/latex], we can conclude that [latex]B\\rightarrow\\sim{M}[\/latex]; that babies cannot manage crocodiles.\r\n\r\nWhile silly, this is a logical conclusion from the given premises.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n<span style=\"background-color: initial; font-size: 0.9em;\"><span style=\"font-size: 1.2em; font-weight: 600; text-transform: uppercase; background-color: initial;\">Law of Detachment (Modus Ponens)<\/span><\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"background-color: initial; font-size: 0.9em;\">The law of detachment applies when a conditional and its antecedent are given as<\/span>\r\n\r\npremises, and the consequent is the conclusion. The general form is:\r\n\r\nPremise: p \u2192 q\r\n\r\nPremise: p\r\n\r\nConclusion: q\r\n\r\nThe Latin name, modus ponens, translates to \u201cmode that affirms\u201d.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Law of Contraposition (Modus Tollens)<\/h3>\r\nThe law of contraposition applies when a conditional and the negation of its consequent\r\nare given as premises, and the negation of its antecedent is the conclusion. The general\r\nform is:\r\nPremise: p \u2192 q\r\nPremise: ~q\r\nConclusion: ~p\r\nThe Latin name, modus tollens, translates to \u201cmode that denies\u201d\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Law of Disjunctive Syllogism<\/h3>\r\nPremise: p V q\r\nPremise:\u00a0~p\r\nConclusion: q\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n\r\n<strong>Forms of Invalid Arguments<\/strong>\r\n\r\n<strong>Fallacy of the Converse<\/strong>\r\nPremise:\u00a0p \u2192 q\r\nPremise: q\r\nConclusion: p\r\n\r\n<strong>Fallacy of the Inverse<\/strong>\r\nPremise:\u00a0p \u2192 q\r\nPremise:\u00a0~p\r\nConclusion:\u00a0~q\r\n\r\n<\/div>\r\n<h2>Logical Inference<\/h2>\r\nSuppose we know that a statement of form [latex]P{\\rightarrow}Q[\/latex] is true. This tells us\u00a0that whenever <em>P<\/em> is true, <em>Q<\/em> will also be true. By itself, [latex]P{\\rightarrow}Q[\/latex]\u00a0being true\u00a0does not tell us that either <em>P<\/em> or <em>Q<\/em> is true (they could both be false, or <em>P<\/em>\u00a0could be false and <em>Q<\/em> true). However if in addition we happen to know\u00a0that <em>P<\/em> is true then it must be that <em>Q<\/em> is true. This is called a <strong>logical\u00a0inference<\/strong>: Given two true statements we can infer that a third statement\u00a0is true. In this instance true statements [latex]P{\\rightarrow}Q[\/latex] and <em>P<\/em> are \u201cadded together\u201d\u00a0to get <em>Q<\/em>. This is described below with [latex]P{\\rightarrow}Q[\/latex]\u00a0stacked one atop the\u00a0other with a line separating them from <em>Q<\/em>. The intended meaning is that [latex]P{\\rightarrow}Q[\/latex]\u00a0combined with <em>P<\/em> produces <em>Q<\/em>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{matrix}P & \\rightarrow & Q\\\\ P & &\\\\ \\hline Q & &\\end{matrix}[\/latex]<\/td>\r\n<td>[latex]\\begin{matrix}P & \\rightarrow & Q\\\\ \\sim Q & &\\\\\\hline \\sim P & &\\end{matrix}[\/latex]<\/td>\r\n<td>[latex]\\begin{matrix}P & \\vee & Q\\\\\\sim P & &\\\\\\hline Q & &\\end{matrix}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTwo other logical inferences are listed above. In each case you should\u00a0convince yourself (based on your knowledge of the relevant truth tables)\u00a0that the truth of the statements above the line forces the statement below\u00a0the line to be true.\r\n\r\nFollowing are some additional useful logical inferences. The first\u00a0expresses the obvious fact that if <em>P<\/em> and <em>Q<\/em> are both true then the statement [latex]P{\\wedge}Q[\/latex] will be true. On the other hand, [latex]P{\\wedge}Q[\/latex]\u00a0being true forces <em>P<\/em> (also <em>Q<\/em>)\u00a0to be true. Finally, if <em>P<\/em> is true, then [latex]P{\\vee}Q[\/latex]\u00a0must be true, no matter what\u00a0statement <em>Q<\/em> is.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{matrix}P & &\\\\Q & &\\\\\\hline P & \\wedge & Q\\end{matrix}[\/latex]<\/td>\r\n<td>[latex]\\begin{matrix} P & \\wedge & Q \\\\\\hline P & & \\end{matrix}[\/latex]<\/td>\r\n<td>[latex]\\begin{matrix} P & & \\\\\\hline P &\\vee &Q\\end{matrix}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">The first two statements in each case are called \u201cpremises\u201d and the final\u00a0statement is the \u201cconclusion.\u201d We combine premises with [latex]{\\wedge}[\/latex] (\u201cand\u201d). The\u00a0premises together imply the conclusion. Thus, the first argument would have [latex]\\left(\\left(P{\\rightarrow}Q\\right){\\wedge}P\\right){\\rightarrow}Q[\/latex]<\/div>\r\n<h2>An Important Note<\/h2>\r\nIt is important to be aware of the reasons that we study logic. There\u00a0are three very significant reasons. First, the truth tables we studied tell\u00a0us the exact meanings of the words such as \u201cand,\u201d \u201cor,\u201d \u201cnot,\u201d and so on.\u00a0For instance, whenever we use or read the \u201cIf..., then\u201d construction in\u00a0a mathematical context, logic tells us exactly what is meant. Second,\u00a0the rules of inference provide a system in which we can produce new\u00a0information (statements) from known information. Finally, logical rules\u00a0such as DeMorgan\u2019s laws help us correctly change certain statements into\u00a0(potentially more useful) statements with the same meaning. Thus logic\u00a0helps us understand the meanings of statements and it also produces new\u00a0meaningful statements.\r\n\r\nLogic is the glue that holds strings of statements together and pins down\u00a0the exact meaning of certain key phrases such as the \u201cIf..., then\u201d or \u201cFor\u00a0all\u201d constructions. Logic is the common language that all mathematicians\u00a0use, so we must have a firm grip on it in order to write and understand\u00a0mathematics.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>What is a logical argument?<\/li>\n<li>Standard Forms of Arguments\n<ul>\n<li>Law of Detachment<\/li>\n<li>Law of Contraposition<\/li>\n<li>Law of Syllogism<\/li>\n<li>Disjunctive Syllogism<\/li>\n<li>Fallacy of the Converse<\/li>\n<li>Fallacy of the Inverse<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>All of the work we have done so far in building the basics of logic has prepared us for the real point which is analyzing logical arguments logically.\u00a0 We want to use logic to evaluate an argument or situation and decide what is and is not reasonable.\u00a0 Setting up our truth tables with letters representing the simple statements allows us to check our opinions and emotions at the door and just focus on the operations.<\/p>\n<p>A logical argument consists of two parts:\u00a0 a set of premises and a conclusion based on the premises.\u00a0 Our goal is to decide whether an argument is valid or invalid.\u00a0 The premises are supporting evidence for the conclusion.\u00a0 Logic is not about deciding if something is true.\u00a0 Logic is about deciding if the conclusion can be deduced by the given premises.<\/p>\n<div class=\"textbox\">\n<h3>Valid Argument<\/h3>\n<p>An argument is <strong>valid<\/strong> if the conclusion follows from the premises.\u00a0 Otherwise, an argument is <strong>invalid<\/strong>.\u00a0 To be <strong>valid<\/strong>, the conclusion of an argument has to follow from the premises.\u00a0 Reasoning that leads to an invalid argument is called a <strong>fallacy<\/strong>.<\/p>\n<\/div>\n<p>The method that we will use in this class to determine the validity of an argument is by using a truth table.<\/p>\n<div class=\"textbox\">\n<h3>Analyzing arguments using truth tables<\/h3>\n<p>To analyze an argument with a truth table:<\/p>\n<ol>\n<li>Represent each of the premises symbolically<\/li>\n<li>Create a conditional statement, joining all the premises with and to form the antecedent, and using the conclusion as the consequent.<\/li>\n<li>Create a truth table for that statement. If it is always true, then the argument is valid.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the argument:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If you bought bread, then you went to the store<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>You bought bread<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>You went to the store<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23681\">Show Solution<\/span><\/p>\n<div id=\"q23681\" class=\"hidden-answer\" style=\"display: none\">\n<p>While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.<\/p>\n<p>We\u2019ll get B represent \u201cyou bought bread\u201d and S represent \u201cyou went to the store\u201d. Then the argument becomes:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]S[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]?<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]B[\/latex]<\/td>\n<td>[latex]S[\/latex]<\/td>\n<td>[latex]B{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left(B{\\rightarrow}S\\right){\\wedge}B[\/latex]<\/td>\n<td>[latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the truth table for [latex]\\left[\\left(B{\\rightarrow}S\\right){\\wedge}B\\right]{\\rightarrow}S[\/latex]\u00a0is always true, this is a valid argument.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the mall, then I\u2019ll buy new jeans.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I buy new jeans, I\u2019ll buy a shirt to go with it.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I got to the mall, I\u2019ll buy a shirt.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Let <em>M<\/em> = I go to the mall, <em>J <\/em>= I buy jeans, and <em>S<\/em> = I buy a shirt.<\/p>\n<p>The premises and conclusion can be stated as:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can construct a truth table for [latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]M[\/latex]<\/td>\n<td>[latex]J[\/latex]<\/td>\n<td>[latex]S[\/latex]<\/td>\n<td>[latex]M{\\rightarrow}J[\/latex]<\/td>\n<td>[latex]J{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)[\/latex]<\/td>\n<td>[latex]M{\\rightarrow}S[\/latex]<\/td>\n<td>[latex]\\left[\\left(M{\\rightarrow}J\\right)\\wedge\\left(J{\\rightarrow}S\\right)\\right]{\\rightarrow}\\left(M{\\rightarrow}S\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the truth table, we can see this is a valid argument.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>Forms of Valid Arguments<\/h3>\n<p>Rather than making a truth table for every argument, we may be able to recognize certain common forms of arguments that are valid (or invalid). If we can determine that an argument fits one of the common forms, we can immediately state whether it is valid or invalid.<\/p>\n<p>Law of Syllogism<\/p>\n<p>Law of Detachment<\/p>\n<p>Law of Contraposition<\/p>\n<p>Law of Disjunctive Syllogism<\/p>\n<\/div>\n<p>The previous problem is an example of a syllogism.<\/p>\n<div class=\"textbox\">\n<h3>Law of Syllogism<\/h3>\n<p>A syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]p{\\rightarrow}q[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]q{\\rightarrow}r[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]p{\\rightarrow}r[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This is sometimes called the transitive property for implication.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>more on the transitive property<\/h3>\n<p>The transitive property appears regularly in the various branches of mathematical study. For example, the transitive property of equality states<\/p>\n<p>if [latex]a = b[\/latex] and [latex]b = c[\/latex] then [latex]a = c[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I work hard, I\u2019ll get a raise.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I get a raise, I\u2019ll buy a boat.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I don\u2019t buy a boat, I must not have worked hard.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q880229\">Show Solution<\/span><\/p>\n<div id=\"q880229\" class=\"hidden-answer\" style=\"display: none\">\n<p>If we let <em>W<\/em> = working hard, <em>R<\/em> = getting a raise, and <em>B<\/em> = buying a boat, then we can represent our argument symbolically:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]H{\\rightarrow}R[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>[latex]R{\\rightarrow}B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>[latex]\\sim{B}{\\rightarrow}{\\sim}H[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We could construct a truth table for this argument, but instead, we will use the notation of the contrapositive we learned earlier to note that the implication [latex]{\\sim}B{\\rightarrow}{\\sim}H[\/latex]\u00a0is equivalent to the implication [latex]H{\\rightarrow}B[\/latex]. Rewritten, we can see that this conclusion is indeed a logical syllogism derived from the premises.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Is this argument valid?<\/p>\n<table>\n<tbody>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the party, I\u2019ll be really tired tomorrow.<\/td>\n<\/tr>\n<tr>\n<td>Premise:<\/td>\n<td>If I go to the party, I\u2019ll get to see friends.<\/td>\n<\/tr>\n<tr>\n<td>Conclusion:<\/td>\n<td>If I don\u2019t see friends, I won\u2019t be tired tomorrow.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><iframe loading=\"lazy\" id=\"ohm25956\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25956&theme=oea&iframe_resize_id=ohm25956&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Lewis Carroll, author of <em>Alice in Wonderland<\/em>, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve the puzzle. In other words, find a logical conclusion from these premises.<\/p>\n<p>All babies are illogical.<\/p>\n<p>Nobody who can manage a crocodile is despised.<\/p>\n<p>Illogical persons are despised.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814448\">Show Solution<\/span><\/p>\n<div id=\"q814448\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.<\/p>\n<p>Then we can write the premises as:<\/p>\n<p style=\"text-align: center;\">[latex]B{\\rightarrow}I\\\\M{\\rightarrow}{\\sim}D\\\\I{\\rightarrow}D[\/latex]<\/p>\n<p>From the first and third premises, we can conclude that [latex]B{\\rightarrow}D[\/latex]; that babies are despised.<\/p>\n<p>Using the contrapositive of the second premised, [latex]D{\\rightarrow}{\\sim}M[\/latex], we can conclude that [latex]B\\rightarrow\\sim{M}[\/latex]; that babies cannot manage crocodiles.<\/p>\n<p>While silly, this is a logical conclusion from the given premises.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<p><span style=\"background-color: initial; font-size: 0.9em;\"><span style=\"font-size: 1.2em; font-weight: 600; text-transform: uppercase; background-color: initial;\">Law of Detachment (Modus Ponens)<\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"background-color: initial; font-size: 0.9em;\">The law of detachment applies when a conditional and its antecedent are given as<\/span><\/p>\n<p>premises, and the consequent is the conclusion. The general form is:<\/p>\n<p>Premise: p \u2192 q<\/p>\n<p>Premise: p<\/p>\n<p>Conclusion: q<\/p>\n<p>The Latin name, modus ponens, translates to \u201cmode that affirms\u201d.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Law of Contraposition (Modus Tollens)<\/h3>\n<p>The law of contraposition applies when a conditional and the negation of its consequent<br \/>\nare given as premises, and the negation of its antecedent is the conclusion. The general<br \/>\nform is:<br \/>\nPremise: p \u2192 q<br \/>\nPremise: ~q<br \/>\nConclusion: ~p<br \/>\nThe Latin name, modus tollens, translates to \u201cmode that denies\u201d<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Law of Disjunctive Syllogism<\/h3>\n<p>Premise: p V q<br \/>\nPremise:\u00a0~p<br \/>\nConclusion: q<\/p>\n<\/div>\n<div class=\"textbox\">\n<p><strong>Forms of Invalid Arguments<\/strong><\/p>\n<p><strong>Fallacy of the Converse<\/strong><br \/>\nPremise:\u00a0p \u2192 q<br \/>\nPremise: q<br \/>\nConclusion: p<\/p>\n<p><strong>Fallacy of the Inverse<\/strong><br \/>\nPremise:\u00a0p \u2192 q<br \/>\nPremise:\u00a0~p<br \/>\nConclusion:\u00a0~q<\/p>\n<\/div>\n<h2>Logical Inference<\/h2>\n<p>Suppose we know that a statement of form [latex]P{\\rightarrow}Q[\/latex] is true. This tells us\u00a0that whenever <em>P<\/em> is true, <em>Q<\/em> will also be true. By itself, [latex]P{\\rightarrow}Q[\/latex]\u00a0being true\u00a0does not tell us that either <em>P<\/em> or <em>Q<\/em> is true (they could both be false, or <em>P<\/em>\u00a0could be false and <em>Q<\/em> true). However if in addition we happen to know\u00a0that <em>P<\/em> is true then it must be that <em>Q<\/em> is true. This is called a <strong>logical\u00a0inference<\/strong>: Given two true statements we can infer that a third statement\u00a0is true. In this instance true statements [latex]P{\\rightarrow}Q[\/latex] and <em>P<\/em> are \u201cadded together\u201d\u00a0to get <em>Q<\/em>. This is described below with [latex]P{\\rightarrow}Q[\/latex]\u00a0stacked one atop the\u00a0other with a line separating them from <em>Q<\/em>. The intended meaning is that [latex]P{\\rightarrow}Q[\/latex]\u00a0combined with <em>P<\/em> produces <em>Q<\/em>.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{matrix}P & \\rightarrow & Q\\\\ P & &\\\\ \\hline Q & &\\end{matrix}[\/latex]<\/td>\n<td>[latex]\\begin{matrix}P & \\rightarrow & Q\\\\ \\sim Q & &\\\\\\hline \\sim P & &\\end{matrix}[\/latex]<\/td>\n<td>[latex]\\begin{matrix}P & \\vee & Q\\\\\\sim P & &\\\\\\hline Q & &\\end{matrix}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Two other logical inferences are listed above. In each case you should\u00a0convince yourself (based on your knowledge of the relevant truth tables)\u00a0that the truth of the statements above the line forces the statement below\u00a0the line to be true.<\/p>\n<p>Following are some additional useful logical inferences. The first\u00a0expresses the obvious fact that if <em>P<\/em> and <em>Q<\/em> are both true then the statement [latex]P{\\wedge}Q[\/latex] will be true. On the other hand, [latex]P{\\wedge}Q[\/latex]\u00a0being true forces <em>P<\/em> (also <em>Q<\/em>)\u00a0to be true. Finally, if <em>P<\/em> is true, then [latex]P{\\vee}Q[\/latex]\u00a0must be true, no matter what\u00a0statement <em>Q<\/em> is.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{matrix}P & &\\\\Q & &\\\\\\hline P & \\wedge & Q\\end{matrix}[\/latex]<\/td>\n<td>[latex]\\begin{matrix} P & \\wedge & Q \\\\\\hline P & & \\end{matrix}[\/latex]<\/td>\n<td>[latex]\\begin{matrix} P & & \\\\\\hline P &\\vee &Q\\end{matrix}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">The first two statements in each case are called \u201cpremises\u201d and the final\u00a0statement is the \u201cconclusion.\u201d We combine premises with [latex]{\\wedge}[\/latex] (\u201cand\u201d). The\u00a0premises together imply the conclusion. Thus, the first argument would have [latex]\\left(\\left(P{\\rightarrow}Q\\right){\\wedge}P\\right){\\rightarrow}Q[\/latex]<\/div>\n<h2>An Important Note<\/h2>\n<p>It is important to be aware of the reasons that we study logic. There\u00a0are three very significant reasons. First, the truth tables we studied tell\u00a0us the exact meanings of the words such as \u201cand,\u201d \u201cor,\u201d \u201cnot,\u201d and so on.\u00a0For instance, whenever we use or read the \u201cIf&#8230;, then\u201d construction in\u00a0a mathematical context, logic tells us exactly what is meant. Second,\u00a0the rules of inference provide a system in which we can produce new\u00a0information (statements) from known information. Finally, logical rules\u00a0such as DeMorgan\u2019s laws help us correctly change certain statements into\u00a0(potentially more useful) statements with the same meaning. Thus logic\u00a0helps us understand the meanings of statements and it also produces new\u00a0meaningful statements.<\/p>\n<p>Logic is the glue that holds strings of statements together and pins down\u00a0the exact meaning of certain key phrases such as the \u201cIf&#8230;, then\u201d or \u201cFor\u00a0all\u201d constructions. Logic is the common language that all mathematicians\u00a0use, so we must have a firm grip on it in order to write and understand\u00a0mathematics.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. 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